What is the integral of ((1+ln(x))^2)/(sqrt(x)) ?

The integral of ((1+ln(x))^2)/(sqrt(x)) is 10sqrt(x)-4sqrt(x)ln(x)+2sqrt(x)ln^2(x)+C

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integral of ((1+ln(x))^2)/(sqrt(x))

\int \frac{( 1 + ln ( x ) ) ^{2}}{x} d x

10 x - 4 x ln (x) + 2 x ln^{2} (x) + C

Solution steps

\int \frac{( 1 + ln ( x ) ) ^{2}}{x} d x

Expand

\frac{( 1 + ln ( x ) ) ^{2}}{x} : \frac{1}{x} + \frac{2 ln ( x )}{x} + \frac{ln ^{2} ( x )}{x}

= \int \frac{1}{x} + \frac{2 ln ( x )}{x} + \frac{ln ^{2} ( x )}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x} d x + \int \frac{2 ln ( x )}{x} d x + \int \frac{ln ^{2} ( x )}{x} d x

\int \frac{1}{x} d x = 2 x

\int \frac{2 ln ( x )}{x} d x = 2 (2 x ln (x) - 4 x)

\int \frac{ln ^{2} ( x )}{x} d x = 2 x ln^{2} (x) - 4 (2 x ln (x) - 4 x)

= 2 x + 2 (2 x ln (x) - 4 x) + 2 x ln^{2} (x) - 4 (2 x ln (x) - 4 x)

Simplify

2 x + 2 (2 x ln (x) - 4 x) + 2 x ln^{2} (x) - 4 (2 x ln (x) - 4 x) : 10 x - 4 x ln (x) + 2 x ln^{2} (x)

= 10 x - 4 x ln (x) + 2 x ln^{2} (x)

Add a constant to the solution

= 10 x - 4 x ln (x) + 2 x ln^{2} (x) + C

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What is the integral of ((1+ln(x))^2)/(sqrt(x)) ?
The integral of ((1+ln(x))^2)/(sqrt(x)) is 10sqrt(x)-4sqrt(x)ln(x)+2sqrt(x)ln^2(x)+C